T850
Mathematics Success – Grade 7
Lesson 33: Relative Frequency
[OBJECTIVE]
The student will use data collected from probability experiments to predict relative
frequency.
[PREREQUISITE SKILLS]
probability, fraction-decimal-percent equivalence
[MATERIALS]
Student pages S434–S443
Fair number cubes (1 per student pair)
Playing cards (1 deck per student pair)
Centimeter cubes (6 red, 4 blue per student pair)
[ESSENTIAL QUESTIONS]
1. Howcanwedefinerelativefrequency?
2. If a coin is tossed 20 times and lands on the head of the coin 8 times, what is the
relativefrequency?
3. Explain how the probability of an event differs from the relative frequency.
[WORDS FOR WORD WALL]
probability, relative frequency, trials, frequency
[GROUPING]
Cooperative Pairs (CP), Whole Group (WG), Individual (I)
*For Cooperative Pairs (CP) activities, assign the roles of Partner A and Partner B to
students. This allows each student to be responsible for designated tasks within the
lesson.
[LEVELS OF TEACHER SUPPORT]
Modeling (M), Guided Practice (GP), Independent Practice (IP)
[MULTIPLE REPRESENTATIONS]
SOLVE, Graphic Organizer, Algebraic Formula, Verbal Description, Concrete Representation,
Pictorial Representation, Table
[WARM-UP] (IP, WG, I) S434 (Answers on T857.)
• Have students turn to S434 in their books to begin the Warm-Up. Students will
work with probability. Monitor students to see if any of them need help during the
Warm-Up. Have students complete the problems and then review the answers as
a class. {Verbal Description}
[HOMEWORK]
Take time to go over the homework from the previous night.
[LESSON] [1 day (1 day = 80 minutes) - M, GP, IP, WG, CP]
Mathematics Success – Grade 7
T851
Lesson 33: Relative Frequency
SOLVE Problem (IP, CP, WG)
S435 (Answers on T858.)
HavestudentsturntoS435intheirbooks.ThefirstproblemisaSOLVEproblem.
Students are going to complete all steps of the SOLVE problem since this is
prior knowledge of determining probability. Have student pairs complete the
SOLVE problem and then go over the answers as a whole group. {SOLVE, Verbal
Description, Graphic Organizer}
Discovery Activity - Extend the SOLVE Problem – Probability and Relative
Frequency
(M, GP, WG, CP, IP) S435, S436, S437, S438
(Answers on T858, T859, T860, T861.)
WG, M, CP, GP:
Have students look at the questions below the SOLVE
problem on S435 in their books. Assign the roles of Partner
A and Partner B. Students will be using the SOLVE problem
from S435 to explore probability. {SOLVE, Verbal Description,
Graphic Organizer, Table, Concrete Representation}
MODELING
Discovery Activity - Extend the SOLVE Problem – Probability and
Relative Frequency
Step 1: Direct students’ attention to the questions below the SOLVE problem.
• HavestudentpairsdiscussQuestions1–2atthebottomofS435then
come back to the whole group discussion to share answers.
• Partner A, how many possibilities were there for the coin toss? (2)
Record.
• PartnerB,whatwastheprobability of tossing the coin and having it
landonheads?( 1 = 0.5 = 50%) Record.
2
• HavestudentsturntoS436.
• PartnerA,doyouthinkthatthecoinwillalwayslandonheadsevery
1outof2times?(No.)Whyorwhynot?(Answersmayvary,butcan
include references to the way the coin is tossed, how it lands, or where
it lands.) Record.
Step 2: Directstudents’attentiontoQuestion4,usingthescenarioaboveit.
• Partner A, what is the probability of tossing a heads on the coin?
(10 = 0.5 = 50%.) Record.
20
• P
artnerB,howdidJasondeterminethis?(Halfofthetossesshouldbe
heads. This would be 10 out of 20 which equals 0.5 or 50%.) Record.
Step 3: UseQuestion5forthestudentstodevelopwhatthetermfrequency means.
• PartnerA,whatoutcomewasJasonhopingforwhenhetossedthecoin?
(heads)Record.Howmanycointosseswereheads?(12)Record.
• Whenwetalkabouthowoftenorhowmanytimessomethinghappens,
we talk about the frequency of that event. For example, if Jason’s
math teacher gives homework assignments 3 days every week, the
frequency of having homework is 3 times per week.
T852
Mathematics Success – Grade 7
Lesson 33: Relative Frequency
• P
artnerB,whatwastheresultofJason’sexperiment?Thecoinlanded
on heads 12 out of 20 tosses. This means that the (frequency), or how
often the coin landed on heads, was 12. Record.
Step 4: If we want to compare the relationship between the frequency of tossing
heads on the coin and how many times the coin was tossed, we need to
know the relationship between the two numbers.
• PartnerB,whatwerethethreeformsweusedtowritetherelationship?
(ratio, decimal, or percent) Record.
• Whenyoudescribehowyouarerelatedtosomeoneinyourfamily,you
say he/she is your (relative). Record.
• WhenwedescribedtherelationshipbetweenhowoftenJasontosseda
head and the total number of events, we described how one value was
related to the other.
• PartnerA,whenwecombinethenumberoftimestheeventoccurred,
(frequency) with the description of how the two values are related,
(relative) we know the (relative frequency)ofJasontossingahead
on the coin. Record.
Step 5: Have students turn to page S437.
• Havepartnersworktogethertocreateadefinitionofrelativefrequency.
(Answers may vary. The ratio or relationship between the number of
favorable events to the total number of events is the relative frequency.
How often something happens divided by the total outcomes is another
definition.)Record.
• PartnerB,describethecomparisonoftheprobabilityandtherelative
frequencyofJason’sproject.(Therelativefrequencywasgreaterthan
the probability.) Record.
Step 6: Have students complete the coin experiment for 30 and then 40 tosses.
• PartnerA,howdidtheprobabilityfor30cointossescomparetothe
relativefrequency?(Answerswillvary.)Record.
• PartnerB,howdidtheprobabilityfor40cointossescomparetothe
relativefrequency?(Answerswillvary.)Record.
• PartnerA,basedontheexperimentsfor30and40cointosses,whatdo
youpredictwouldhappenwith50cointosses?(Therelativefrequency
would be closer to the probability.) Record.
• PartnerB,basedontheexperimentsfor30and40cointosses,whatdo
youpredictwouldhappenwith100cointosses?(Therelativefrequency
would be closer to the probability.) Record.
Mathematics Success – Grade 7
T853
Lesson 33: Relative Frequency
IP, CP, WG:
Have students work in student pairs to complete the
SOLVE problem on S438. They will be using relative
frequency to complete the problem. Then come back
together as a class and share their results. {Verbal
Description, Graphic Organizer, SOLVE}
Relative Frequency and Probability with Cards
(M, GP, WG, CP, IP)
S439, S440 (Answers on T862, T863.)
WG, M, CP, GP:
Have students turn to page S439 in their books.
Assign the roles of Partner A and Partner B. Pass
out a deck of cards to each student pair. Students
will be using cards to determine relative frequency
and compare it with probability of events.
{Verbal Description, Graphic Organizer, Concrete
Representation}
MODELING
Relative Frequency and Probability with Cards
Step 1: Direct students’ attention to the top of S439. Pass out a deck of cards for
each student pair to take out a 2, 3, 4, 5, and 6. Students will use the
number cards to complete the questions in the graphic organizer.
1
• PartnerA,whatistheprobabilityofchoosinga2? 5 Record.
• PartnerB,explaintheprobabilityof2.(Thereisonlyonecardwitha
2. There are a total of 5 cards, so the probability is 5 = 0.2 = 20%.)
• Have students complete the probability column for Problems 2 – 5.
Review answers as a whole group.
1
Step 2: Nowdirectstudents’attentiontotheRelativeFrequencycolumn.
• Havestudentstake5cards(2–6)fromthenumbercardsandplace
them face down on their desk. Have students complete 10 trials for
each of the 5 probabilities and mark the number of times that a number
occurredinthechart,thenfillintherelativefrequencycolumn.Record
your answer each time.
T854
Mathematics Success – Grade 7
Lesson 33: Relative Frequency
Step 3: HavestudentslookonS440toanswerQuestions1–4andsharehowthe
probability differed from the relative frequency.
• Havestudentpairsdiscusswhythismayhavehappened.(Withthe
relative frequency there are variables such as choosing the card. There
were only 10 trials. Other answers are possible.) Record.
• Did each student pair have the same relative frequency? (Answers
will vary. Probably not, because of the differences in the experimental
process.) Record.
Step 4: Have student pairs discuss the meaning of the following terms and write
theirdefinitions.Bepreparedtoshareyourexplanationwiththewhole
group.
• Probability:(howlikelysomethingistohappen)Record.
• Relative Frequency: (observed number of successful events or
outcomes for a number of trials.) Record.
IP, CP, WG:
Have students work in pairs and go back to S439 to
complete the graphic organizer for trials of 20 and then
answerQuestions5–8onS440.Aftercompletingthechart
and questions, come back together as a class to share
answers. {Verbal Description, Graphic Organizer}
Conclusions about Relative Frequency and Probability
WG, M, CP, GP:
(M, GP, WG, CP, IP)
S441 (Answers on T864.)
Have students turn to page S441 in their books. Assign the
roles of Partner A and Partner B. Students will be applying
what they have learned about the relationship between
probability and relative frequency. {Verbal Description,
Graphic Organizer}
Mathematics Success – Grade 7
T855
Lesson 33: Relative Frequency
MODELING
Conclusions about Relative Frequency and Probability
Step 1: Direct students’ attention to the top of S441. The graphic organizer
contains one situation on coin tosses, which the teacher and student will
complete together, and one situation on fair number cubes which the
students will complete as a student pair.
• PartnerA,identifytheeventinSituation1.(cointoss200times)
• Partner B, what was the outcome of the coin toss? (108 heads, 92
tails)
• PartnerA,whatistheprobabilityofflippingaheadonthecoin?
50
1
(100 = 2 = 50%) Record in the graphic organizer.
• P
artnerB,whatwastherelativefrequencyofheadsinthesituation?
108
(200 = 54%)
Record in the graphic organizer.
Step 2: Direct students’ attention to Situation 2.
• PartnerA,identifytheeventinSituation2.(cointoss120times)
• P
artnerB,whatwastheoutcomeofthecointoss?(50heads,70tails)
• PartnerA,whatistheprobabilityofflippingaheadonthecoin?
50
1
(100 = 2 = 50%) Record in the graphic organizer.
• PartnerB,whatwastherelativefrequencyofheadsinthesituation?
( 50 = 41.67%) Record in the graphic organizer.
120
Step 3: Directstudents’attentiontoQuestion3.
• PartnerA,whichsituationhadarelativefrequencythatwascloserto
theprobabilityof50%?(thecointosswith200trials)Record.
• Havestudentpairsdiscussthenextquestionandbepreparedtoshare
their predictions.
• PartnerB,canyoumakeapredictionwhythatwastheresult?(The
situation with more trials has a smaller difference between the
probability and the relative frequency.) Record.
• Partner A, based on this information, what do you predict would
happen if the experiment had 500 trials? (The difference between
the probability and the relative frequency would be smaller than the
situation with 200 trials and closer to 50%.) Record.
IP, CP, WG:
avestudentsworkinpairstocompleteQuestions4–6
H
on the graphic organizer. Then come back together as a
class to share their answers. {Verbal Description, Graphic
Organizer}